In revising the "Lessons on Higher Algebra" various minor changes have been made in the articles of the first edition, and a number of articles added. A section on Determinants has also been introduced. In explanation of the brevity of these Lessons, it ought to be said, that the mathematical course for which they were written includes further work for more advanced students, in which the text-book used is Burnside and Panton's "Theory of Equations." The presence, in an algebra book, of so unusual a feature as a section on ...
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In revising the "Lessons on Higher Algebra" various minor changes have been made in the articles of the first edition, and a number of articles added. A section on Determinants has also been introduced. In explanation of the brevity of these Lessons, it ought to be said, that the mathematical course for which they were written includes further work for more advanced students, in which the text-book used is Burnside and Panton's "Theory of Equations." The presence, in an algebra book, of so unusual a feature as a section on the nature of mathematical reasoning, calls for more than a passing word. It cannot have escaped the notice of those who are acquainted with the prevailing methods of instruction in mathematics and logic that, on the one hand, the average teacher of mathematics gives little or no attention to the nature of the processes according to which mathematical reasoning is conducted; while, on the other hand, the teacher of logic almost ignores mathematics as a source of illustrations of the principles with which he is dealing. A preface is not the place to discuss the causes or effects of this divorce of two subjects which might go hand in hand; but I must express my conviction that when we discover those methods of instruction which are essential to sound education, they will be found to include logic for the student of mathematics, and mathematics for the student of logic. Since writing the Appendix I have received the Report of the Committee on Secondary School Studies, appointed at the meeting of the National Educational Association, July 9, 1892, including the Conference Report of the Committee on Mathematics, of which Professor Simon Newcomb, of the Johns Hopkins University, is chairman. This Conference Report confirms me in the belief that any successful effort to improve the teaching of mathematics, both in colleges and secondary schools, must recognize the claims of logic. "The very fact that demonstrative geometry is the most elaborate illustration of formal logic in the entire curriculum of the student, makes the consideration of these elementary principles of logic more interesting and profitable in this connection than in any other." (Report: Mathematics, p. 115.) The analytical branch of mathematics, however, scarcely falls behind the geometrical in affording 'elaborate illustration of formal logic'; and the identity of the logic of algebra and the logic of geometry is probably best shown by putting algebraic and geometrical illustrations side by side....
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